## Advances in Differential Equations

- Adv. Differential Equations
- Volume 11, Number 4 (2006), 361-398.

### Positive solutions to singular semilinear elliptic equations with critical potential on cone-like domains

Vitali Liskevich, Sofya Lyakhova, and Vitaly Moroz

#### Abstract

We study the existence and nonexistence of positive (super-) solutions to a singular semilinear elliptic equation $$-\nabla\cdot(|x|^A\nabla u)-B|x|^{A-2}u=C|x|^{A-\sigma}u^p$$ in cone--like domains of $\mathbb R^N$ ($N\ge 2$), for the full range of parameters $A,B,\sigma,p\in\mathbb R$ and $C>0$. We provide a characterization of the set of $(p,\sigma)\in\mathbb R^2$ such that the equation has no positive (super-),solutions, depending on the values of $A,B$ and the principal Dirichlet eigenvalue of the cross--section of the cone. The proofs are based on the explicit construction of appropriate barriers and involve the analysis of asymptotic behavior of super-harmonic functions associated to the Laplace operator with critical potentials, Phragmén-Lindelöf type comparison arguments and an improved version of Hardy's inequality in cone--like domains.

#### Article information

**Source**

Adv. Differential Equations, Volume 11, Number 4 (2006), 361-398.

**Dates**

First available in Project Euclid: 18 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1355867701

**Mathematical Reviews number (MathSciNet)**

MR2215620

**Zentralblatt MATH identifier**

1194.35170

**Subjects**

Primary: 35J60: Nonlinear elliptic equations

Secondary: 35B05: Oscillation, zeros of solutions, mean value theorems, etc. 35B33: Critical exponents

#### Citation

Liskevich, Vitali; Lyakhova, Sofya; Moroz, Vitaly. Positive solutions to singular semilinear elliptic equations with critical potential on cone-like domains. Adv. Differential Equations 11 (2006), no. 4, 361--398. https://projecteuclid.org/euclid.ade/1355867701